This question is from $\text{p-adic numbers}.$
My questions are-
$(1)$ Is the set $ \ (\mathbb{Z}_p \setminus \mathbb{Z}) \cap \mathbb{Q}$ non-empty? If non-empty, then what are the elements or the intersection set?
$(2)$ Is the set $ \ (\mathbb{Z}_p \setminus \mathbb{Z}) \cap \mathbb{Q}_p $ non-empty? If non-empty what are the elements or the intersection set?
I can not conclude the answer.
Please someone help me with details answer or at least hints.
Hint: $\frac{1}{2} \in \mathbb{Z}_3 \setminus \mathbb{Z}$, since $$\frac{1}{2}= \frac{-1}{1-3} = -1-3-9-27- \dots.$$ That (1) is nonempty implies that (2) is.
Similarly, you can prove that any $\frac{a}{b} \in \mathbb{Z}_3 \setminus \mathbb{Z}$ so long $3 \nmid b.$